ISL 2022 G2

In the acute-angled triangle $A B C$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $A F$. The lines through $P$ parallel to $A C$ and $A B$ meet $B C$ at $D$ and $E$, respectively. Points $X \neq A$ and $Y \neq A$ lie on the circles $A B D$ and $A C E$, respectively, such that $D A=D X$ and $E A=E Y$. Prove that $B, C, X$ and $Y$ are concyclic. (Netherlands)